Beyond the mean field with a multiparticle-multihole wave function and the Gogny force

Beyond the mean field with a

multiparticle-multihole wave function

and the Gogny force

N.Pillet

J.-F.Berger

M.Girod

CEA Bruyères-le-Châtel

E.Caurier

Ires Strasbourg

E

E

Nuclear Correlations

Pairing correlations (BCS-HFB)

Correlations associated to collective oscillations

Small amplitude (RPA)

Large amplitude (GCM)

(non conservation of particle number )

(Pauli principle not respected )

Aim of our work

An unified treatment of the correlations beyond the mean field

•conserving the particle number

•enforcing the Pauli principle

•using the Gogny interaction

Description of the pairing-type correlations in all pairing regimes

Will the D1S force be adapted to describe correlations beyond the mean field in this approach ?

Description of particle-vibration coupling

Description of collective and non collective states

Trial wave function

Superposition of Slater determinants corresponding to multiparticle-multihole excitations upon a given ground state of HF type

Similar to the m-scheme

Simultaneous Excitations of protons and neutrons

{d+n} are axially deformed harmonic oscillator

statesDescription of the nucleus in a deformed basis

Some Properties of the mpmh wave function

• Importance of the different ph excitation orders ?

• Treatment of the proton-neutron residual part of the interaction

• The projected BCS wave function on particle number is a subset of the mpmh wave function

specific ph excitations (pair excitations)

specific mixing coefficients (particle coefficients x hole coefficients)

Richardson exact solution of the Pairing hamiltonian

Picket fence model

(for one type of particle)

g

The exact solution corresponds to the multiparticle-multihole wave function including all the configurations built as pair excitations

Test of the importance of the different terms in the mpmh wave function expansion : presently pairing-type correlations (2p2h, 4p4h ...)

εi

εi+1

d

R.W. Richardson, Phys.Rev. 141 (1966) 949

N.Pillet, N.Sandulescu, Nguyen Van Giai and J.-F.Berger , Phys.Rev. C71 , 044306 (2005)

Ground state Correlation energy

gc=0.24

ΔEcorr(BCS)~ 20%

Ecorr=E(g≠0)-E(g=0)

Ground state

Occupation probabilities

R.W. Richardson, Phys.Rev. 141 (1966) 949

Picket fence model

Variational Principle

Determination of • the mixing coefficients

• the optimized single particle states used in building the

Slater determinants.Definition

Total energy

One-body density

Minimization of the energy functional

Correlation energy

Hamiltonian ijkl

kljiij

ji aaaaklVij4

1aajKiH

Determination of the mixing coefficients

Use of the Shell Model technology !

Using Wick’s theorem, one can extract the usual mean field part and the residual part

VHHH

h1 h2p1 p2

p1 p2 h2h1

h1 p3p1

p2 p1 h3h2

h1

h1

h2

p1

p2 p1

p2

h2

h1

h4

h3p2

p1 p3

p4

h2

h1

|n-m|=2

|n-m|=1

|n-m|=0

npnh< Φτ |:V:|Φτ>mpmh

Determination of optimized single particle states

Use of the mean field technology !

•Iterative resolution → selfconsistent procedure

•No inert core

•Shift of single particle states with respect to those of the HF solution

In the general case, h and ρ are no longer diagonal

simultaneously

Preliminary results with the D1S Gogny force in the case of pairing-type correlations

Ground state, β=0

(without self-consistency)

-Ecor (BCS) =0.124 MeV

-TrΔΚ ~ 2.1 MeV

Nsh = 9 Nsh = 9

T(0,0)= 89.87%

T(0,1)= 7.50%

T(0,2)= 0.24%

T(2,0)= 0.03%

T(1,1)= 0.17%

T(1,0)= 2.19%

T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.0003%

Ground state, β=0

(without self-consistency)

-Ecor (BCS) =0.588 MeV

-TrΔΚ ~ 6.7 MeV

Nsh = 9Nsh = 9

T(0,0)= 82.65%

T(0,1)= 10.02%

T(0,2)= 0.56%

T(0,2)= 0.23%

T(1,1)= 0.54%

T(1,0)= 5.98%

T(3,0) + T(0,3) + T(2,1) + T(1,2) = 0.03% ~ 15 keV

Occupation probabilities (without self-consistency)

Occupation probabilities (without self-consistency)

Outlook

•the effect of the selfconsistency

•more general correlations than the pairing-type ones

•connection with RPA

•excited states

•axially deformed nuclei

•e-e, e-o, o-o nuclei

•charge radii, bulk properties

.........

Two particles-two levels model

εa= 0

εα= ε

BCS

mpmh

Ground state, β=0

(without self-consistency)

-Ecor (BCS) =0.588 MeV

-TrΔΚ ~ 2.1 MeV

-Ecor (BCS) =0.588 MeV

-TrΔΚ ~ 6.7 MeV

Ground state, β=0

(without self-consistency)

Numerical application

0.375 0.146 0.625 0.854

0.450 0.379 0.550 0.578

0.488 0.422 0.512 0.578

Projected BCS wave function (PBCS) on particle number

BCS wave function

Notation

PBCS : • contains particular ph excitations

• specific mixing coefficients : particle coefficients x hole coefficients

Ground state Correlation energy

Rearrangement terms

•Polarization effect